Contravariant finiteness and iterated strong tilting

Let $\mathcal{P}^{<\infty} (\Lambda$-mod$)$ be the category of finitely generated left modules of finite projective dimension over a basic Artin algebra $\Lambda$. We develop an applicable criterion that reduces the test for contravariant finiteness of $\mathcal{P}^{<\infty} (\Lambda$ -mod$)$ in $\Lambda$-mod to corner algebras $e \Lambda e$ for suitable idempotents $e \in \Lambda$. The reduction substantially facilitates access to the numerous homological benefits entailed by contravariant finiteness of $\mathcal{P}^{<\infty} (\Lambda$-mod$)$. The consequences pursued hinge on the fact that this finiteness condition is known to be equivalent to the existence of a strong tilting object in $\Lambda$-mod. We characterize the situation in which the process of strongly tilting $\Lambda$-mod allows for arbitrary iteration: This occurs precisely when, in the strongly tilted module category mod-$\widetilde{\Lambda}$, the subcategory of modules of finite projective dimension is in turn contravariantly finite; the latter can, once again, be tested on suitable corners $e \Lambda e$ of the original algebra $\Lambda$. In the (frequently occurring) positive case, the sequence of consecutive strong tilts, $\widetilde{\Lambda}$, $ \widetilde{\widetilde{\Lambda}}$, $\widetilde{\widetilde{\widetilde{\Lambda}}}, \dots$, is shown to be periodic with period $2$ (up to Morita equivalence); moreover, any two adjacent categories in the sequence $\mathcal{P}^{<\infty} ( $mod-$\widetilde{\Lambda})$, $\mathcal{P}^{<\infty}(\widetilde{\widetilde{\Lambda}}-mod)$, $\mathcal{P}^{<\infty}($ mod-$\widetilde{\widetilde{\widetilde{\Lambda}}}), \dots$ are dual via contravariant Hom-functors induced by tilting bimodules which are strong on both sides.


Introduction
Co-and contravariant finiteness of a subcategory A of the category Λ-mod of finitely generated modules over an Artin algebra Λ were first considered by Auslander and Smalø in [2] and [3]: If A = add(A) is closed under extensions, the combination of these two finiteness conditions implies the existence of internal almost split sequences in A. Here we focus on the full subcategory A = P <∞ (Λ-mod) consisting of the modules of finite projective dimension; in this situation contravariant finiteness implies the dual property, covariant finiteness [15]. (See Section 2 for notation and definitions of the italicized terms.) Subsequently, it was found that contravariant finiteness of P <∞ (Λ-mod) entails a plethora of additional homological benefits for Λ-mod and the unrestricted module category Λ-Mod. Namely, this finiteness condition not only validates the finitistic dimension conjectures for left Λ-modules, i.e., confirms that l.findim(Λ) = l.Findim(Λ) < ∞ in that case, but gives rise to an intrinsic description of the Λ-modules of finite projective dimension; see [1] and [15].
Crucial to our present investigation are the following facts: P <∞ (Λ-mod) is contravariantly finite in Λ-mod if and only if Λ-mod contains a strong tilting module, i.e., a tilting module T which is relatively Ext-injective within the category P <∞ (Λ-mod). Such a module T is alternatively characterized by the condition that the contravariant functor Hom Λ (−, T ) induces a duality between P <∞ (Λ-mod) and a specifiable resolving subcategory of P <∞ (mod-Λ), whereΛ = End Λ (T ) op (see [13] and [14]). In this situation we say that mod -Λ results from Λ-mod via strong tilting. Finally, we recall that, up to isomorphism, Λ-mod has at most one basic strong tilting module; see [1].
In the present article we tackle the foremost obstacle that stands in the way of applying the theory we sketched: namely, the notoriously difficult task of deciding whether, for specific choices of Λ, the category P <∞ (Λ-mod) is contravariantly finite in Λ-mod. The driving goals of our investigation are to significantly reduce this difficulty and to explore the possibility and effect of iterating the process of strongly tilting Λ-mod. In both directions we make headway, next to retrieving known results in a simpler, more uniform format.
In more detail: In the first of our main results we address an Artin algebra Λ for which P <∞ (Λ-mod) is known to be contravariantly finite in Λ-mod. In general, P <∞ (mod-Λ) need not inherit contravariant finiteness. However, whenever it does, the process of strongly tilting Λ-mod may be iterated in the following sense: Λ-mod allows for unlimited iteration of strong tilting if there exists an infinite sequence of basic algebras (Λ i ) i≥0 with Λ 0 = Λ such that mod -Λ i+1 results from Λ i -mod via strong tilting if i is even, and Λ i+1 -mod results from mod -Λ i via strong tilting if i is odd.
Theorem A. (For a more complete version, see Theorem 3.1 and Corollary 3. 4

.)
Suppose that Λ is a basic Artin algebra such that P <∞ (Λ-mod) is contravariantly finite in Λ-mod. Let Λ T be the corresponding basic strong tilting module andΛ = End Λ (T ) op .
To return to the problem of determining the contravariant finiteness status of P <∞ (Λ-mod) in the first place, we suppose that e 1 , . . . , e n form a complete set of orthogonal primitive idempotents of Λ and that the subsum e = e 1 + ⋅ ⋅ ⋅ + e m includes all idempotents e i that give rise to simple left modules of infinite projective dimension. Our second and third main results refer to this setting.
Theorem B. (For a complete version, see Theorem 4.6.) Adopt the notation of the preceding paragraph, and suppose that eΛ(1 − e) has finite projective dimension as a left module over the corner algebra eΛe. Then P <∞ (Λ-mod) is contravariantly finite in Λ-mod if and only if P <∞ (eΛe-mod) is contravariantly finite in eΛe-mod.
Theorem 4.6 also spells out how, in case of existence, the minimal (right) P <∞ (Λ-mod)-approximation of M ∈ Λ-mod relates to the minimal P <∞ (eΛemod)approximation of eM . The ensuing possibility of shucking off primitive idempotents in checking for contravariant finiteness of P <∞ (Λ-mod) not only provides effortless access to most of the cases in which this property has previously been established (formerly involving considerable effort), but yields contravariant finiteness in far more general situations.
Our argument relies on an exploration of the torsion-torsionfree triple (C, T , F ) which is associated to the idempotent e, and on the Giraud subcategory G of Λ-Mod (in the sense of Gabriel [9]) corresponding to the hereditary torsion pair (T , F ).
In order to extend Theorem B to an efficient test of whether the stronger conclusions of Theorem A hold for Λ, we once more assume that Λ is a basic Artin algebra such that P <∞ (Λ-mod) is contravariantly finite. As before, we let e 1 , . . . , e n ∈ Λ be a complete set of primitive idempotents, T ∈ Λ-mod the basic strong tilting module, andΛ = End Λ (T ) op . By S i we denote the simple left Λ-module corresponding to e i . It is well-known that the indecomposable direct summands T 1 , . . . , T n of Λ T coincide, up to isomorphism, with the distinct indecomposable direct summands of ⊕ 1≤i≤n A i , where A i is the minimal P <∞ (Λ-mod)-approximation of the indecomposable injective left module with socle S i . Our analysis of the T i will pave the road towards showing that the question of unlimited iterability of strong tilting of Λ-mod can in turn be played back to the corner algebra eΛe for any idempotent e as specified in Theorem B.
Theorem C. Let m ≤ n be chosen such that pdim Λ S j < ∞ for j > m. Set e = e 1 + ⋯ + e m and assume that pdim eΛe eΛ(1 − e) < ∞. Then: (1) (Proposition 5.2.) The number of distinct indecomposable direct summands of ⊕ 1≤i≤m A i is m. Denote them by T 1 , . . . , T m and set T ′ = ⊕ 1≤i≤m T i .
For j ≥ m + 1, the approximation A j of S j decomposes in the form A j = T j ⊕ U j , where T j is indecomposable with the property that S j is the only simple module of finite projective dimension in soc T j , and all indecomposable direct summands of U j occur as direct summands of T ′ .
(2) (For more detail, see Theorem 5.4, and Corollaries 5.5, 5.6.) Λ-mod allows for unlimited iteration of strong tilting if and only if the same is true for eΛe-mod.
Our proof of Theorem C is based on connections between T ∈ Λ-mod and the strong tilting objects in eΛe-mod and in the Giraud subcategory G.
Earlier results for truncated path algebras (i.e., for algebras of the form KQ I, where K is a field, Q a quiver, and I the ideal generated by the paths of length L+1 for some L ≥ 1), showcase the effects of iterated strong tilting and the associated dualities among the P <∞ -categories encountered along the iterations [13]; these findings readily follow from the above theorems. In fact, the present results yield substantial generalizations of the picture that arose in the truncated case (Propositions 6.1, 6.2, Corollaries 6.3 and 6.4). From our reduction technique it also follows that, for any left serial algebra Λ, both P <∞ (Λ-mod) and P <∞ (mod-Λ) are contravariantly finite in Λ-mod and mod-Λ, respectively (the former fact had been known; see [6]). Further applications address Artin algebras arising from Morita contexts, such as algebras of triangular matrix type (Theorem 6.6, Corollary 6.7, Examples 6.9), and the elimination of simple modules of low projective dimension in the test for contravariant finiteness (Proposition 5.7). Section 2 assembles minimal conceptual background and builds the tools required for proving our main theorems. Section 3 provides a general characterization of the situation in which Λ-mod allows for unlimited iteration of strong tilting. The results targeting tests for contravariant finiteness of Λ-mod and for the availability of repeated strong tilts of Λ-mod are contained in Sections 4 and 5, respectively. In Section 6, we specify applications.

Notation, background and auxiliaries
Throughout, Λ will be a basic Artin algebra, and J its Jacobson radical. We point out that the restriction to basic algebras does not affect the generality of our investigation; we adopt it for increased transparency of the underlying ideas. Λ-Mod and Λ-mod stand for the categories of all (resp., all finitely generated) left Λ-modules. Further, S 1 , . . . , S n will be isomorphism representatives of the simple objects in Λ-Mod, and top M , soc M will stand for the top and socle of M ∈ Λ-Mod, respectively. By P <∞ (Λ-Mod) (resp., P <∞ (Λ-mod)) we denote the full subcategory of Λ-Mod (resp., Λ-mod) having as objects the modules of finite projective dimension. Moreover, for any finitely generated Λ-module M , we denote by add(M ) the full subcategory of Λ-mod consisting of the direct summands of finite direct sums of copies of M . The module M is basic if it has no indecomposable direct summands of multiplicity > 1.
2.1. Contravariant finiteness of P <∞ (Λ-mod) and strong tilting modules. Following Miyashita [18], we call a left Λ-module T a tilting module in case (1) T belongs to P <∞ (Λ-mod), (2) Ext i Λ (T, T ) = 0 for i ≥ 1, and (3) there exists an exact sequence 0 → Λ Λ → T 0 → ⋯ → T m → 0 with T j ∈ add(T ). It is well known that any basic tilting module has precisely n = rank K 0 (Λ) indecomposable direct summands, and that any tilting module T ∈ Λ-mod gives rise to a left-right symmetric situation as follows: IfΛ = End Λ (T ) op , then the rightΛ-module TΛ is in turn a tilting module and EndΛ(T ) is canonically isomorphic to Λ. This justifies the reference to a tilting bimodule Λ TΛ. Strong tilting modules were first considered by Auslander and Reiten in [1]. We introduce them via a characterization equivalent to the original definition (see [1]): Namely, we call a tilting module T strong if it satisfies the following relative injectivity condition in P <∞ (Λ-mod): (4) Ext i (M, T ) = 0 for all M ∈ P <∞ (Λ-mod) and i ≥ 1. It was shown by Auslander and Reiten [loc.cit.] that Λ-mod has a strong tilting module if and only if the category P <∞ (Λ-mod) is contravariantly finite in Λ-mod, a property which we will recall next. Moreover, according to [loc.cit], in case of existence, there is precisely one basic strong tilting module in Λ-mod, up to isomorphism.
The concept of contravariant finiteness of subcategories of Λ-mod has its roots in work of Auslander and Smalø [2]: Namely, the category P <∞ (Λ-mod) is contravariantly finite in Λ-mod provided that for each object M ∈ Λ-mod, the functor Hom Λ (−, M ) P <∞ (Λ-mod) is finitely generated in the category of additive contravariant functors P <∞ (Λ-mod) → Ab. This condition translates into the following requirement for arbitrary M ∈ Λ-mod: There is an object A ∈ P <∞ (Λ-mod), together with a map φ ∈ Hom Λ (A, M ), such that each map in Hom Λ (P <∞ (Λ-mod), M ) factors through φ. Any such pair (A, φ) is called a (right ) P <∞ (Λ-mod)-approximation of M . Since we will only consider right approximations, we will frequently omit the qualifier "right". By a mild abuse of terminology, we will moreover refer to the domain A of φ as a P <∞ (Λ-mod)-approximation of M . Suppose M has a P <∞ (Λ-mod)-approximation, say φ ∶ A → M . As was shown by Auslander-Smalø in [loc.cit.], up to isomorphism, there is only one approximation φ ∶ A(M ) → M such that A(M ) has minimal length. It is alternatively characterized by the condition that any endomorphism u of A(M ) which satisfies φ ○ u = φ is an automorphism; we say that the map φ is (right) minimal if this implication holds. Since it is unlikely to cause misunderstandings, we will also refer to A(M ) as "the" minimal P <∞ (Λ-mod)-approximation of M whenever convenient.
The mentioned existence result by Auslander-Reiten will be crucial in the sequel. We state it for easy reference.
There exists a strong tilting module T ∈ Λ-mod if and only if P <∞ (Λ-mod) is contravariantly finite in Λ-mod. In the positive case, the basic strong tilting module is unique up to isomorphism: it is the direct sum of the distinct indecomposable modules C ∈ P <∞ (Λ-mod) which satisfy Ext i Λ (P <∞ (Λ-mod), C) = 0 for i ≥ 1.
(2) [13, Supplement II in Section 2.A] A more explicit description of the indecomposable direct summands of a strong tilting module T , when it exists, can be obtained from the following: Let A be the minimal P <∞ (Λ-mod)-approximation of the minimal injective cogenerator of Λ-Mod. Then add(T ) = add(A).

2.2.
The TTF-triple associated to an idempotent e in Λ. We fix an idempotent element e ∈ Λ. By Jans [17] (see also [20,Section VI.8]), e defines a TTF triple (C e , T e , F e ) in the category Λ-Mod; this means that the pairs (C e , T e ) and (T e , F e ) are both torsion pairs. The torsion, resp. torsionfree, classes are as follows: (a) C e consists of the Λ-modules C generated by Λe, i.e., the modules of the form C = ΛeC. (b) T e consists of the left Λ-modules annihilated by e. (c) F e consists of the Λ-modules F with the property that the annihilator ann F (eΛ) of eΛ in F is zero. Observe that the torsion pair (T e , F e ) is hereditary, whence the corresponding torsion radical is left exact (see [20,Proposition VI.3.1]). On the other hand, (C e , T e ) fails to be hereditary in general.
Further notation: Since we will keep the idempotent e fixed, we will more briefly write The torsion radicals ∇ and ∆ associated to the pairs (C, T ) and (T , F ) are the idempotent subfunctors of the identity functor on Λ-Mod given by: A third functor Λ-Mod → Λ-Mod will serve to render the constructions in Sections 4 and 5 more transparent. It assigns to each Λ-module M the following subfactor of M : If M is finitely generated, the core of M has maximal length among the subfactors V U of M such that the top and socle of V U belong to F , i.e., such that top(V U ), soc(V U ) ∈ add(Λe Je). In fact, it can easily be seen that this maximality property determines core(M ) up to isomorphism. Observe moreover, that C ∩ F consists of the Λ-modules which coincide with their cores.
We point out that, in general, the core of a module may also have simple composition factors annihilated by e; for a broader spectrum of examples, see [13], [14].
2.3. The adjoint pair (Λe ⊗ eΛe −, e). By e we denote the functor Λ-Mod → eΛe-Mod which sends M to eM . The two functors of the title thus play the role of induction and restriction in the exchange of information between eΛe-modules on one hand and Λ-modules on the other. Accordingly, Λe ⊗ eΛe − ∶ eΛe-Mod → Λ-Mod is left adjoint to e.
We briefly explore the functor "restriction followed by induction", Λ-Mod → Λ-Mod, which sends M ∈ Λ-Mod to M ‡ ∶= Λe ⊗ eΛe eM . Evidently, this functor sees only the core of a Λ-module M ; indeed, M ‡ is naturally isomorphic to core(M ) ‡ . On the other hand, it preserves this core, as the following lemma shows. More precisely, the counit ǫ of the adjunction induces a family of isomorphisms Proof. Clearly, ǫ gives rise to a family of epimorphisms and Ker(ρ M ) contains ∆(Λe ⊗ eΛe eM ). That, conversely, Ker(ρ M ) is contained in ∆(Λe⊗ eΛe eM ) follows from the facts that e(ρ M ) ∶ eΛe⊗ eΛe eM → e core(M ) = eM is an isomorphism in eΛe-mod and ∆(Λe ⊗ eΛe eM ) is the largest submodule of Λe ⊗ eΛe eM which is annihilated by e. We conclude that the maps ǫ M are indeed isomorphisms.
2.4. The Giraud subcategory of Λ-Mod corresponding to the torsion pair (T , F ). We apply well-known facts about localization with respect to a hereditary torsion class T to the specialized situation where T = T e . We refer the reader to [10] and [20, Chapter IX] for detail.
Recall that the Giraud subcategory G of Λ-Mod relative to the hereditary torsion pair (T , F ) is a realization, inside Λ-Mod, of the quotient category Λ-Mod T ; here we identify the torsion class T with the (full) localizing subcategory of Λ-Mod that has object class T . In particular, G is a Grothendieck category. Concretely, G is the full subcategory of Λ-Mod whose objects are the torsionfree Λ-modules F with the following restricted injectivity property: Ext 1 Λ (X, F ) = 0 for all (cyclic) torsion modules X ∈ T .
It is well known that the fully faithful inclusion functor ι ∶ G → Λ-Mod has an exact left adjoint σ ∶ Λ-Mod → G, which is identifiable with the quotient functor Λ-Mod → Λ-Mod T ; in particular, the pair (G, σ) has the universal property of such a quotient. We will write M σ for σ(M ). In parallel, To The explicit description of σ ∶ Λ-Mod → Λ-Mod reveals that this functor, in turn, sees only the core of a Λ-module M, i.e., M σ = core(M ) σ , and that it preserves cores, meaning that core(M σ ) is canonically isomorphic to core(M ); in fact, M σ is an essential extension of core(M ) which is maximal relative to the requirement that this core be preserved. The following alternative incarnations of the category G will be helpful in Sections 4 and 5. 1. [20, Proposition XI.8.6] The categories G and eΛe-Mod are equivalent. Quasiinverse equivalences send F ∈ G to eF in one direction, and send X ∈ eΛe-Mod to (Λe ⊗ eΛe X) σ in the other.
The indecomposable projective objects of G are (Λe i ) σ for 1 ≤ i ≤ m, and the indecomposable injectives are E(S i ) σ = E(S i ) for 1 ≤ i ≤ m.

[9]
The subcategory C ∩ F of Λ-Mod is in turn equivalent to G. Quasi-inverse equivalences send M = core(M ) to M σ ; in reverse, F ∈ G is sent to core(F ) = ∇(F ). In particular, the functors σ and σ ○ core from Λ-Mod to Λ-Mod are naturally isomorphic.
The indecomposable projective objects of C ∩ F are core(Λe i ) = Λe i ∆(Λe i ) for 1 ≤ i ≤ m, and the indecomposable injectives are core E(S i ) ) = ∇ E(S i ) for 1 ≤ i ≤ m.
We add some notation: The unit of the adjunction (σ, ι) is the natural transformation If M is torsionfree, we identify µ M with the inclusion map M ↪ M σ ⊆ E(M ). Clearly, µ M is an isomorphism precisely when M belongs to G. Remark 2.5. We point out that the mentioned functors linking the subcategories A of Λ-Mod (resp. of eΛe-Mod) introduced in subsections 2.2-2.4 restrict to functors connecting the pertinent intersections A ∩ Λ-mod, resp., A ∩ eΛe-mod and A ∩ G.
Example 2.6. (Return to Example 2.2.) In this instance, (I j ) σ = I j for j = 1, 2, because I 1 and I 2 are injective objects of F , and consequently of G. Now let j ∈ {3, 4}. In either case, I j ∆(I j ) ≅ S 2 ⊕S 2 , whence E j ∶= E(I j ∆(I j )) ≅ I 2 ⊕I 2 . Since the torsion submodule of (I 2 ⊕ I 2 ) (S 2 ⊕ S 2 ) is zero, we obtain (I j ) σ = S 2 ⊕ S 2 , and the map µ Ij ∶ I j → (I j ) σ = S 2 ⊕ S 2 is the obvious projection.
2.5. The poset of essential ∆-extensions of a morphism. In constructing P <∞ (Λ-mod)-approximations of Λ-modules M from P <∞ (eΛe-mod)-approximations of eM , passage to maximal extensions of the type described in this subsection will be crucial. Throughout we refer to the torsion theory (T , F ) introduced in 2.2.

Definition.
Given a morphism f ∶ M → Y in Λ-mod, we consider the following eligible extensions of f : These are the pairs (L, g), where L is an essential extension of M with the additional property that L M ∈ T and g ∈ Hom Λ (L, Y ) satisfies We say that two eligible pairs (L, g) and The set E f of all essential ∆-extensions [(L, g)] of f is a poset under the following partial order: We comment on the legitimacy of the final definition: Welldefinedness of the relation ⪯ is clear. To check that it is antisymmetric, it suffices to observe that the existence of monomorphisms L → L ′ and L ′ → L forces the finitely generated Λ-modules L and L ′ to have the same length, whence monomorphisms between them are isomorphisms.
Proposition 2.7. Suppose that M and Y are finitely generated torsionfree left Λmodules, and let f ∈ Hom Λ (M, Y ). Then the poset (E f , ⪯) has a maximum. To address our first claim, we note that the lengths of increasing sequences in E f are bounded from above by the composition length of E(M ). In particular, each element of E f is majorized by a maximal element in this set. Therefore, we only need to prove that any two elements To prove the final claim, we note that the map χ in the pullback diagram is an injection since µ Y is. Thus the submodule µ −1 M (χ(N )) of M σ is isomorphic to N . On replacing N by this copy and adjusting φ and χ accordingly, we obtain another Since ψ extends to an automorphism of E(M ), we do not lose generality in viewing ψ as a set inclusion. We thus obtain a commutative diagram, in which κ denotes the inclusion map N ′ ↪ M σ : In particular, v is a monomorphism. We infer that N and N ′ have the same length and conclude that ψ is an isomorphism. This proves

Iteration of strong tilting
When can the process of strongly tilting Λ-mod to a category of right modules, mod-Λ, be iterated? In the positive case, how do the resulting sequences of strongly tilted module categories behave? The main purpose of this section is to answer these questions. More specifically, we will show that, whenever Λ-mod can be strongly tilted twice, the process can be iterated arbitrarily and turns periodic after the initial step. Roughly speaking: In case P <∞ (mod-Λ) is in turn contravariantly finite in mod-Λ, the initial transition from Λ toΛ increases the homological symmetry by increasing the number of simple modules of finite projective dimension. This symmetrization makes the subsequent sequence mod-Λ ↝Λ-mod ↝ ⋯ periodic with period 2.
The statement of the following theorem is based on part (1) of Theorem 2.1.
Theorem 3.1. Let Λ be a basic Artin algebra such that P <∞ (Λ-mod) is contravariantly finite in Λ-mod. Moreover, let Λ T be the corresponding basic strong tilting module, andΛ = End Λ (T ) op . Suppose that P <∞ (mod-Λ) is in turn contravariantly finite in mod-Λ, thus giving rise to a basic tilting bimoduleΛTΛ which is strong in ThenT is strong on both sides. In particular, the process of strongly tilting Λ-mod then allows for unlimited iteration, yielding a sequence of basic Artin algebras Λ = Λ 0 , Λ 1 , Λ 2 , Λ 3 , . . . with the property that Λ i and Λ j are isomorphic whenever i and j are positive integers with the same parity.
Moreover, the algebras Λ 0 and Λ 2 are isomorphic precisely when the tilting bimodule Λ TΛ is strong on both sides, i.e., when T =T . In light of the hypothesis that Λ T is a strong tilting module, we thus find that all simple rightΛ-modules are contained in soc TΛ, up to isomorphism. Moreover, we deduce that strongness ofΛT as a tilting object inΛ-mod will follow if we can show that all simple rightΛ-modules embed intoTΛ.
To realize such an embedding, let (Λ J )Λ =S 1 ⊕ ⋯ ⊕S n , where theS i are simple. Further denote by E(S 1 ⊕ ⋯ ⊕S n ) an injective envelope, and letÃ be a minimal P <∞ (mod-Λ)-approximation of the latter. In light of our hypothesis thatTΛ is a strong tilting object in mod-Λ, the categories add(TΛ) and add(Ã) coincide; see Theorem 2.1 (2). Embed TΛ into an injective rightΛ-module, say Then f is injective, and we deduce that all simple rightΛ-modules embed intoÃ m . Clearly, this implies that all simples in mod-Λ embed into socÃ. Invoking the fact that add(Ã) = add(TΛ), we conclude that allS i occur in socTΛ as well, which proves two-sided strongness ofΛTΛ.
The final assertion follows from the preceding argument.
The result is sharp in the sense that the existence of a strong tilting module in Λ-mod does not, by itself, provide sufficient homological symmetry to allow for iteration of strong tilting. [16]. Let Λ = KQ I, where Q is the quiver
For a first application, we combine Theorem 3.1 with the fact that strong tilting modules induce contravariant equivalences of categories of modules of finite projective dimension (see [13, Reference Theorem III and Theorem 1] or [14,Theorems 7,8]). This yields if and only if Λ TΛ is two-sided strong, i.e., precisely when T ≅T .
In case the process of strongly tilting Λ-mod permits for iteration, periodicity of the sequence Λ-mod ↝ mod-Λ ↝ ⋯ may set in with delay. In fact, if Λ is a truncated path algebra one always obtains a sequence of basic strong tilts, Λ-mod ↝ mod-Λ ↝ ⋯ (see [13,Theorem 19]); for this class of algebras, Λ-mod ≈Λ-mod if and only if Q does not have a precyclic source (see [7,Corollary 7.2]). We refer to [13] and [14] for examples based on quivers with precyclic sources where the transition from the tilting module TΛ to the basic strong tilting moduleTΛ in mod-Λ is displayed.
We conclude the section with a variant of the criterion of Auslander and Green for a strong tilting module T ∈ Λ-mod to be strong also as a tilting module over End Λ (T ) op ; it was implicitly used in the proof of Theorem 3.1. The upcoming version of the criterion does not rely on structural information regarding Λ T and provides additional motivation for the detection of twosided strongness.
Observation 3.5. Suppose Λ is an Artin algebra such that P <∞ (Λ-mod) is contravariantly finite in Λ-mod. Then the following statements are equivalent: (2) Every simple left Λ-module of infinite projective dimension embeds into a Λmodule of finite projective dimension.
Proof. "(1) ⇒ (2)" is immediate from the criterion of Auslander and Green quoted in the proof of Theorem 3.1. "(2) ⇒ (1)": By (2), every simple left Λ-module S embeds into a module M = M (S) of finite projective dimension. Consider an embedding ι ∶ M ↪ E of M into an injective module E, and let φ ∶ A → E be a minimal P <∞ (Λ-mod)-approximation. Then ι factors through φ, which shows M to be contained in A up to isomorphism. Hence so is S. Since add(A) ⊆ add(T ) by hypothesis (cf. the proof of Theorem 3.1), we conclude that S embeds into T . Thus an application of [1, Proposition 6.5] yields (1).
The equivalence "(1) ⇐⇒ (3) Our main objective in this section is to show that, in exploring whether the subcategory P <∞ (Λ-mod) is contravariantly finite in Λ-mod, a comparatively mild hypothesis (4.1(ii) below) permits us to eliminate primitive idempotents which correspond to simple left Λ-modules of finite projective dimension. This reduces the problem to a more manageable corner eΛe of Λ.

The setting.
Setting 4.1 (Blanket hypotheses). Throughout this section, we will assume that the idempotent e ∈ Λ satisfies the following two conditions: With regard to 4.1(ii), we mention that, in case Λ is a path algebra modulo relations, the quiver and relations of eΛe are available from those of Λ, making this condition computationally accessible. In the present setting, the adjoint pair (Λe ⊗ eΛe −, e) is particularly useful towards the transfer of homological information between the categories Λ-mod and eΛe-mod. Namely: Proof. By [8, Lemma 1.2(b)], the functor e ∶ Λ-mod → eΛe-mod preserves finite projective dimension. In light of the canonical isomorphism e ○ (Λe ⊗ eΛe −) ≅ id eΛe-mod this shows that Λe ⊗ eΛe − reflects finite projective dimension.
To see that Λe⊗ eΛe − preserves finite projective dimension, let X ∈ P <∞ (eΛe-mod), and let be a projective resolution in eΛe-mod. Then all terms Λe ⊗ eΛe Q j of the complex Λe ⊗ eΛe X are projective over Λ; indeed, the Q j belong to add( eΛe eΛe), whence the modules Λe ⊗ eΛe Q j belong to the category add(Λe ⊗ eΛe eΛe) = add( Λ Λe). Moreover, all homology modules of Λe ⊗ eΛe X have finite projective dimension in Λ-mod; indeed, due to the natural isomorphism e(Λe ⊗ eΛe X) ≅ X, they are all annihilated by e, whence their simple composition factors are direct summands of Λ(1 − e) J(1 − e). Writing (F i ) for the differential (Λe ⊗ eΛe f i ) of Λe ⊗ eΛe X, we find that Ker(F m ) has finite projective dimension. In view of projectivity of Λe ⊗ eΛe Q m , so does Im(F m ), whence we obtain finiteness of pdim Λ Ker(F m−1 ) from that of pdim Λ Ker(F m−1 ) Im(F m ). An obvious induction thus shows that Im(F 0 ) ≅ Λ ⊗ eΛe X has finite projective dimension over Λ.
That the functor e reflects finite projective dimension is now seen as follows: Given M ∈ Λ-mod such that e(M ) has finite projective dimension over eΛe, we apply the conclusion of the preceding paragraph and Lemma 2.3 to deduce that pdim Λ core(M ) < ∞. Once more invoking 4.1(i), we conclude that pdim Λ M < ∞. Let M ∈ Λ-mod, suppose that eM has a P <∞ (eΛe-mod)-approximation, and let q ∶ X → eM be a minimal such approximation.
Proof. Referring to the explicit category equivalences of Lemma 2.4, we see that (Λe ⊗ eΛe q) σ is a minimal approximation of M σ with respect to the subcategory In light of the preceding remarks, the domain of this map has finite projective dimension also in Λ-mod. Right minimality of (Λe ⊗ eΛe q) σ in Λ-mod is clear. To see that (Λe ⊗ eΛe q) σ is actually a P <∞ (Λ-mod)approximation of M σ , let U ∈ P <∞ (Λ-mod) and f ∈ Hom Λ (U, M σ ). Since M σ is torsionfree and U ∆(U ) again belongs to P <∞ (Λ-mod), we do not lose generality in assuming that U is torsionfree. Consequently, µ U ∶ U → U σ is an embedding with the property that U σ U ∈ T . Now the restricted injectivity property of M σ yields an extension of f to f * ∶ U σ → M σ , i.e., f = f * ○ µ U . Since U σ has finite projective dimension in G ∩(Λ-mod), we further obtain a factorization f * = (Λe ⊗ eΛe q) σ ○g for some g ∈ Hom Λ (U σ , (Λe ⊗ eΛe X) σ ). This clearly yields a factorization of f through (Λe ⊗ eΛe q) σ .
The next observation depends only on Condition 4.1(i). If pdim N < ∞ and ρ is an epimorphism, then ρ is an isomorphism. In particular: Proof. For the first implication it suffices to observe that, under the given premise, τ ∶ N → M is in turn a P <∞ (Λ-mod)-approximation of M . Minimality of p thus shows that length(A(M )) ≤ length(N ). Therefore the epimorphism ρ is an isomorphism.
To derive the final implication, assume that M is torsionfree, and apply the preceding implication to the canonical map ρ ∶ A(M ) → A(M ) ∆ A(M ) .

4.2.
The main theorem. The upcoming theorem shows that, in the situation of 4.1, contravariant finiteness of P <∞ (Λ-mod) in Λ-mod is equivalent to contravariant finiteness of P <∞ (eΛe-mod) in eΛe-mod. In fact, we obtain sharper information, relating minimal P <∞ -approximations in Λ-mod to minimal P <∞ -approximations in eΛe-mod.
As before, (T , F ) = (T e , F e ) is the torsion pair of subsection 2.2. Recall that the class T consists of the Λ-modules all of whose composition factors belong to add Λ(1 −e) J(1 −e) , while F consists of the modules F with soc F ∈ add(Λe Je). Then P <∞ (Λ-mod) is contravariantly finite in Λ-mod if and only if P <∞ (eΛe-mod) is contravariantly finite in eΛe-mod.
In more detail: The following two implications hold for every finitely generated left Λ-module F ∈ F .
(1) If p ∶ M → F is a minimal right P <∞ (Λ-mod)-approximation of F , then p eM ∶ eM → eF is a minimal right P <∞ (eΛe-mod)-approximation of eF .
(2) Suppose that q ∶ X → eF is a minimal right P <∞ (eΛemod)-approximation of eF , and let f be the composition   (2) of the theorem is naturally isomorphic to the map core(1 ⊗ q) ∶ core(Λe ⊗ eΛe X) → core(F ) and, modulo the canonical isomorphism core(F ) ≅ Λe⊗eΛeeF ∆(Λe⊗eΛeeF ) , the map ǫ F ∶ core(F ) = ∇(F ) → F is the embedding of core(F ) into F . Proof. We start by showing that the first assertion follows from (1) and (2). Indeed, the correspondence S ↔ eS is a bijection between the isomorphism classes of simple Λ-modules in add(Λe Je) and those of the simple left eΛe-modules. In light of [1,Proposition 3.7] and the fact that the simples in add Λ(1 − e) J(1 − e) have finite projective dimension, it thus suffices to show that any simple module S ∈ add(Λe Je) has a P <∞ (Λ-mod)-approximation precisely when eS has a P <∞ (eΛe-mod)-approximation. Clearly, the simple summands S of Λe Je satisfy S = ΛeS ∈ F , and thus the required equivalence arises from (1) and (2) as a special case.
To prove that p ′ is (right) minimal, it suffices to show that no nonzero direct summand of eM is contained in the kernel of p ′ . So suppose eM = X ⊕ Y with p ′ (Y ) = 0, and let π ∶ eM → X be the projection onto X along Y . The latter gives rise to a corresponding projection pr ∶ M ‡ ∆(M ‡ ) → X † ∆(X † ). Bearing in mind that M ∈ F by Lemma 4.5, we obtain the following commutative diagram with exact rows where the left-hand square is the pushout of ǫ M and pr. In particular, ρ is an epimorphism and u is a monomorphism. Both of the flanking terms of the bottom exact sequence have finite projective dimension: For X † ∆(X † ) this follows from Lemma 4.3, for the righthand term it follows from Coker(u) ≅ Coker(ǫ M ) ≅ M ∇(M ). Consequently, also N has finite projective dimension. On defining q to be the restriction of ǫ F ○ (1 ⊗ p ′ ) to X † ∆(X † ), one readily checks that the following diagram commutes: The universal property of the pushout therefore yields a map v ∈ Hom Λ (N, F ) such that v ○ ρ = p. By Lemma 4.5, ρ is an isomorphism, whence so is pr, meaning that Y † ∆(Y † ) = 0. Thus Y † ∈ T , and we conclude that Y ≅ e(Y † ) = 0. This confirms minimality of p ′ .
(2) Let q ∶ X → eF and f be as in the assertion, and suppose that the pair (M, p) represents the maximum of the set E f . To check that p ∶ M → F is a P <∞ (Λ-mod)-approximation of F , let h ∈ Hom Λ (N, F ) with N ∈ P <∞ (Λ-mod). Since ∆(F ) = 0 and N ∆(N ) again belongs to P <∞ (Λ-mod), we may assume that ∆(N ) = 0. In light of Lemma 4.3, the restriction h ′ = h eN ∶ eN → eF is a morphism in Hom eΛe P <∞ (eΛe-mod), eF , whence it factors through q, say h ′ = q ○ η with η ∈ Hom eΛe (eN, X). Again referring to Lemma 4.3, we moreover see that the map 1 ⊗ η ∶ N ‡ ∆(N ‡ ) → X † ∆(X † ) induced by η is a morphism in P <∞ (Λ-mod). In order to suitably extend 1 ⊗ η to a homomorphism with domain N via the monomorphism ǫ N ∶ N ‡ ∆(N ‡ ) → N , we consider the pushout diagram of the pair (ǫ N , 1 ⊗ η), as shown in the diagram below: Since the image of ǫ N is ∇(N ), we may use the argumentation in the proof of (1) to infer that N ′ ∈ P <∞ (Λ-mod). By the naturality of ǫ (see Lemma 2.3), ǫ F ○ 1 ⊗ h ′ coincides with h ○ ǫ N , which yields f ○ (1 ⊗ η) = h ○ ǫ N ; in other words, the above diagram fully commutes. Thus the universal property of the pushout provides us with a map φ ∈ Hom Λ (N ′ , F ) such that φ ○ λ = f and φ ○ g = h. Given that F is torsionfree, φ factors through the canonical map π ∶ N ′ → N ′ ∆(N ′ ); denote the induced map N ′ ∆(N ′ ) → F by φ, and note that the composition λ ∶= π ○ λ is still an embedding, since Im(λ) ∩ ∆(N ′ ) = 0; it is thus harmless to view λ as a set inclusion. We will ascertain that the pair N ′ ∆(N ′ ), φ gives rise to a class in E f : Indeed, the cokernel of λ belongs to T , since it is an epimorphic image of N ∇(N ). Torsionfreeness of N ′ ∆(N ′ ) moreover guarantees that λ is an essential extension. That φ extends f , is immediate from our construction. The inequality It is straightforward to deduce that h factors through p as required.
To see that p is a right minimal morphism, consider a decomposition p = p 1 0 ∶ M = U ⊕ V → F , where p 1 is right minimal. Due to the fact that [(M, p)] ∈ E f , the domain M of p is an essential extension of X † ∆(X † ) such that the quotient of M modulo X † ∆(X † ) belongs to T . This means that eM = e X † ∆(X † ) , whence the induced map p ′ = p eM ∶ eM → eF coincides with the restriction f ′ of f to e X † ∆(X † ) . In light of the natural equivalence e ○ (Λe ⊗ eΛe −) ≅ 1 eΛe-mod , this identifies f ′ with the minimal approximation q ∶ X → eF . Due to the matrix decomposition p ′ = p ′ 1 0 ∶ X ≅ eU ⊕ eV → eF , where p ′ 1 = p 1 eU , right minimality of q thus yields eV = 0, so that V ∈ T . On the other hand, V ∈ F ; this follows from the fact that M ∈ F , because the torsionfree module X † ∆(X † ) is an essential submodule of M . Consequently, V = 0, which proves minimality of p as claimed.
We apply Theorem 4.6 to the situation where the P <∞ -categories of eΛe-mod and (equivalently) Λ-mod are contravariantly finite in the corresponding ambient module categories. The proposition below picks up the theme of Lemma 4.4 and Remark 4.7. It reinforces our understanding of the links among the minimal approximations of objects M ∈ Λ-mod and those of the corresponding Λ-modules M ∆(M ), M σ , and core(M ). These connections underlie the upcoming exploration of the basic algebraΛ that results from strongly tilting Λ-mod.
We conclude that q ′ coincides with p up to isomorphism, whence A(M ) ∆(A(M )) ≅ A(M ) and ∆ A(M ) ≅ ∆(M ). In particular, the composition has the postulated properties. Remark 4.9. Analyzing the proof of the preceding proposition, one observes the following: If π X ∶ X → X = X ∆(X) and p X ∶ A(X) → X denote the canonical projection and the minimal P <∞ (Λ-mod)-approximation of a module X, then one has a composition of pullbacks In applying Theorem 4.6, we typically decompose e and 1 − e into primitive idempotents: e = e 1 + ⋯ + e m and 1 − e = e m+1 + ⋯ + e n . Since part (2) of Theorem 4.6 aims at reducing the contravariant finiteness test for P <∞ (Λ-mod) to that for P <∞ (eΛe-mod), we are interested in making eΛe as "small" as possible. Hence the situation where all simple left modules S i of finite projective dimension correspond to idempotents e i for i ≥ m + 1 is of particular interest. A modification of an example by Fuller-Saorín (see [8,Example 4.2]) shows that, even for monomial algebras Λ, neither of the conditions 4.1(i) nor 4.1(ii) in the hypothesis of Theorem 4.6 is dispensable.
which is defined by the graphs of its indecomposable projective left Λ-modules,  This shows that the left eΛe-module eΛ(1 − e) has infinite projective dimension. To see that the equivalence of Theorem 4.6 fails, observe that l.findim eΛe = 0, whence P <∞ (eΛe-mod) is contravariantly finite in eΛe-mod. Yet P <∞ (Λ-mod) is not contravariantly finite in Λ-mod, since the simple module S 1 does not have a P <∞ (Λ-mod)-approximation. Indeed, consider the family (M n ) n∈N of objects in P <∞ (Λ-mod) shown below: With the aid of Criterion 10 of [11], one readily checks that no homomorphism φ ∈ Hom Λ P <∞ (Λ-mod), S 1 allows for factorization of all of the following maps f n ∈ Hom Λ (M n , S 1 ) in the form f n = φ ○ g n ; here f n (x 1 ) = e 1 ∈ Λe 1 Je 1 and f n (x i ) = 0 for i > 1.
We mention that, in the presence of condition 4.1(ii), condition 4.1(i) in the hypothesis of Theorem 4.6 is not superfluous either. Instances attesting to this are ubiquitous. In the above example, take e = e 1 , and note that this choice makes eΛe semisimple.

5.
The basic strong tilting object in P <∞ (Λ-mod) and its endomorphism algebra In this section, we focus on the situation where P <∞ (Λ-mod) is contravariantly finite in Λ-mod. LetΛ = End Λ (T ) op , where Λ T is the basic strong tilting object in Λ-mod. The guiding question is this: When is P <∞ (mod-Λ) in turn contravariantly finite in mod-Λ? In light of Theorem 3.1, this amounts to the question of when the process of strongly tilting Λ-mod can be repeated arbitrarily. As witnessed by Example 3.3, the possibility of iteration is not automatic in case Λ-mod can be strongly tilted once. By Theorem 4.6, P <∞ (Λ-mod) is contravariantly finite in Λ-mod in this setting. We will introduce an idempotentẽ inΛ which naturally corresponds to e. Our objective is to show that the blanket hypotheses 4.1 of the previous section carry over to mod-Λ, meaning that the rightΛ-module (1 −ẽ)Λ (1 −ẽ)J in turn has finite projective dimension, and the corner (1 −ẽ)Λẽ has finite projective dimension as a rightẽΛẽ-module. In light of Theorem 4.6, this will then allow us to deduce contravariant finiteness of P <∞ (mod-Λ) from that of P <∞ (mod-ẽΛẽ).
To this end, we will first assemble some information about the Λ-module structure of the basic strong tilting module Λ T .
From Section 2.1 we know that add T = add A, where A is a minimal P <∞ (Λ-mod)approximation of an injective cogenerator of Λ-mod. In particular, any such minimal approximation A is a strong tilting module. To pin down a candidate for A, we decompose both e and 1 − e into sums of primitive idempotents of Λ, say e = e 1 + ⋯ + e m and 1 − e = e m+1 + ⋯ + e n . Moreover, we let S i = Λe i Je i be the corresponding simple modules, and choose A i to be the minimal P <∞ (Λ-mod)approximations of their injective envelopes E(S i ). Clearly, A ∶= ⊕ 1≤i≤n A i is then as required. Since the injective objects of the Giraud subcategory G of Λ-mod relative to the torsion pair (T , F ) (see 2.4) are precisely the torsionfree injective Λ-modules, the subsum ⊕ 1≤i≤m E(S i ) is an injective cogenerator in G. Moreover, according to Lemma 2.2, left multiplication by e induces an equivalence of categories G ∩ Λ-mod ≅ (eΛe)-mod, and ⊕ 1≤i≤m eE(S i ) is an injective cogenerator in eΛe-mod. We separately explore the direct sums ⊕ 1≤i≤m A i and ⊕ m+1≤i≤n A i . Proposition 5.2. Assume the hypotheses 5.1, and let T ∈ Λ-mod be the basic strong tilting module. Then add(⊕ 1≤i≤m A i ) contains precisely m isomorphism classes of indecomposable modules, represented by T 1 , . . . , T m say. If T ′ = ⊕ 1≤i≤m T i , then the direct summand T ′ of T is an object of G, and eT ′ is the basic strong tilting object in eΛe-mod, up to isomorphism.
Proof. By Theorem 4.6(1), ⊕ 1≤i≤m eA i is a P <∞ (eΛe-mod)-approximation of the injective cogenerator ⊕ 1≤i≤m eE(S i ) in eΛe-mod, which shows that ⊕ 1≤i≤m eA i is a strong tilting object in eΛe-mod. Since the rank of K 0 (eΛe) equals m, this implies that add(⊕ 1≤i≤m eA i ) contains precisely m isomorphism classes of indecomposable eΛe-modules (see Section 2.1). According to Proposition 4.8, the sum ⊕ 1≤i≤m A i in turn belongs to G, and therefore the category equivalence G ∩Λ-mod ≅ eΛe-mod guarantees that add(⊕ 1≤i≤m A i ) contains the same number of isomorphism classes of indecomposable objects. If these are represented by T 1 , . . . , T m , then e(⊕ 1≤i≤m T i ) is the basic strong tilting eΛe-module by construction. This proves the first claim.
For the final claims, let i ≥ m+1. We observe that S i = soc E(S i ) is the only simple torsion module which embeds into A i ; indeed, this is immediate from the fact that ∆(A i ) ≅ ∆ E(S i ) by Proposition 4.8(c). Hence precisely one of the indecomposable direct summands of A i contains a copy of S i in its socle; say A i = T i ⊕ U i with T i indecomposable and S i = ∆(soc T i ), while ∆(U i ) = 0. Then ⊕ m+1≤i≤n T i consists of n−m pairwise nonisomorphic direct summands in add(A), none of which occurs among T 1 , . . . , T m , and we conclude that T = ⊕ 1≤i≤n T i up to isomorphism. Since the U i are torsionfree and hence do not belong to add(⊕ m+1≤i≤n T i ), they belong to add(⊕ 1≤i≤m T i ).
In view of the fact that T ′ is torsionfree, the final claim follows.
Definition. We refer to Proposition 5.2. Letẽ i ∈Λ be the projection T → T i ⊆ T with respect to the decomposition T = ⊕ 1≤i≤n T i . Viewingẽ i as an endomorphism of T , we thus obtain a primitive idempotent inΛ. We defineẽ ∶= ∑ 1≤i≤mẽi ∈Λ. Thusẽ is the projection T → T ′ ⊆ T along T ′′ ∶= ⊕ m+1≤i≤n T i . A j , we need to identify the minimal P <∞ (Λ mod)-approximation of (I j ) σ = S 2 ⊕ S 2 (see Example 2.6). Then, by Lemma 4.4, such an approximation is of the form (1⊗q) σ ⊕(1⊗q) σ , where q ∶ X → eS 2 is the minimal P <∞ (eΛe mod)-approximation, whence the projective cover eΛe 2 → eS 2 since eΛe is self-injective. It easily follows that (1 ⊗ q) σ gets identified with the canonical projection ρ ∶ (Λe 2 ) σ = I 1 → S 2 . By Proposition 4.8 and Remark 4.9 again, the Λ-module A j is the upper left corner of the pullback of ρ⊕ρ ∶ I 1 ⊕I 1 → S 2 ⊕S 2 and the projection µ Ij ∶ I j → (I j ) σ = S 2 ⊕S 2 . It then follows that T 3 = A 3 is given by the following diagram, and T 4 = A 4 is obtained from T 3 by factoring out the copy of S 3 in the socle of T 3 .
op is immediate from Proposition 5.2 and the comments that precede it. As above, suppose that e = e 1 + ⋯ + e m and 1 − e = e m+1 + ⋯ + e n , where the e i are primitive. Regarding condition (i) of the claim: By strongness of Λ T , the functor Hom Λ (−, T ) ∶ Λ-mod → mod-Λ induces a contravariant equivalence between P <∞ (Λ-mod) and a certain subcategory of P <∞ (mod-Λ), whence pdimΛ Hom Λ (S i , T ) < ∞ for all i ≥ m + 1 (see [13, Reference Theorem III and Theorem 1] or [14,Theorems 7,8]). So we only need to show that the rightΛ-module Hom Λ (S i , T ) is isomorphic tõ e iΛ ẽ iJ for m + 1 ≤ i ≤ n. Let i ≥ m + 1. In light of Proposition 5.2, S i = ∆(soc T i ) is the only occurrence of S i in the socle of T , and therefore any homomorphism S i → T maps S i onto ∆(soc T i ). If in i ∶ S i ↪ T i ⊆ T is an embedding, we thus find that Hom Λ (S i , T ) ≅ in iΛ . One checks that the latter module is annihilated by the radicalJ ofΛ, but not annihilated byẽ i , and concludes that Hom Λ (S i , T ) ≅ẽ iΛ ẽ iJ as postulated.
Theorem 5.4 ensures that, with a "duplicate" inΛ of the original idempotent e ∈ Λ, the test provided by Theorem 4.6 is again available towards deciding whether P <∞ (mod-Λ) is contravariantly finite in mod-Λ.
In light of the fact that the basic strong tilt eΛe of eΛe coincides withẽΛẽ, we thus obtain: Not only is existence of a strong tilting object in Λ-mod equivalent to existence of a strong tilting object in eΛe-mod under conditions 4.1, but the same hypothesis implies that Λ-mod allows for arbitrary repetitions of strong tilting precisely when this is true for eΛe-mod. This conclusion compiles information from Theorems 3.1, 4.6 and Corollary 5.5. A first straightforward application of our techniques shows that, in testing for contravariant finiteness of P <∞ (Λ-mod) or iterability of strong tilting, we may automatically discard the idempotents corresponding to the simple left Λ-modules of projective dimension at most 1. Namely: Proposition 5.7. Let Λ be a basic Artin algebra on which we fix the complete set of primitive idempotents {e 1 , ..., e r , e r+1 , ..., e n }, ordered in such a way that pdim(Λe i Je i ) ≤ 1 for i > r. For e = ∑ r i=1 e i , the following assertions are equivalent: (1) P <∞ (Λ-mod) is contravariantly finite in Λ-mod (resp., Λ-mod allows for unlimited iteration of strong tilting); (2) P <∞ (eΛe-mod) is contravariantly finite in eΛe-mod (resp., eΛe-mod allows for unlimited iteration of strong tilting).

Applications and examples
By way of the equivalences established in the previous sections, we can now easily secure contravariant finiteness of P <∞ (Λ-mod) and P <∞ (mod-Λ) in cases in which this originally required a considerable effort. Moreover, the unifying reasons behind these results become more transparent through the reduction and permit us to expand the settings to which they apply. 6.1. Precyclic/postcyclic vertices and normed Loewy lengths. An instance in which the simplification gained by reduction to corner algebras stands out is that of truncated path algebras and their strong tilts (see [13] and [14]); without a reduction technique, it is challenging to confirm that Λ-mod allows for unlimited iteration of strong tilting in this case.
In light of Theorem 4.6, the first step, namely to confirm contravariant finiteness of P <∞ (Λ-mod) for truncated Λ, has now been trivialized. To generalize it, recall that, given any path algebra modulo relations, Λ = KQ I, a vertex e i of Q (systematically identified with a primitive idempotent of Λ) is called precyclic if there exists a path of length ≥ 0 which starts in e i and ends on an oriented cycle; the attribute postcyclic is dual, and e i is called critical if it is both pre-and postcyclic. Clearly, all vertices which give rise to simple modules of infinite projective dimension are among the precyclic ones; moreover e i Λe j = 0 whenever e i is precyclic, but e j is not. Theorem 4.6 thus yields Proposition 6.1. Let Λ = KQ I be an arbitrary path algebra modulo relations,and let e be the sum of the primitive idempotents corresponding to the precyclic vertices of Q. If P <∞ (eΛe-mod) is contravariantly finite in eΛe-mod, e.g., if l.findim eΛe = 0, then P <∞ (Λ-mod) is contravariantly finite in Λ-mod.
For truncated Λ, the final condition concerning the left finitistic dimension of eΛe is clearly satisfied since all indecomposable projective left eΛe-modules eΛe i for precyclic e i have the same Loewy length; this is not necessarily true for the indecomposable injective left eΛe-modules, but it is for those whose socles correspond to critical vertices; namely, if e ′ is the sum of the critical vertices of Q, then the indecomposable injective left e ′ Λe ′ -modules also have coinciding Loewy lengths in the truncated case. The combination of these two conditions which norm the Loewy lengths of certain projective or injective modules is, in fact, all that is needed to guarantee that Λ-mod allows for iterated strong tilting. Suppose that all indecomposable projective left eΛe-modules have the same Loewy length, and that the analogous equality holds for the Loewy lengths of the indecomposable injective left e ′ Λe ′ -modules. Then Λ-mod allows for unlimited iteration of strong tilting, thus giving rise to a sequence of related module categories Λ-mod ↝ mod-Λ ↝Λ-mod ↝ ⋯. Starting with the first strong tilt, mod-Λ, the Morita equivalence classes of these categories repeat periodically with period 2.
As we already pointed out above, the conditions (i) and (ii) of Setting 4.1 are satisfied for the pair (Λ-mod, e), whence, by Theorem 5.4, we only need to show that eΛe-mod allows for unlimited iteration of strong tilting. Set Λ ′ = eΛe, and let Q ′ , resp. J ′ , be the quiver and Jacobson radical of Λ ′ , respectively. We already know that the category P <∞ (Λ ′ -mod) is contravariantly finite in Λ ′ -mod, due to the vanishing of the left finitistic dimension of Λ ′ . The latter in fact entails that the basic strong left Λ ′ -module T ′ is a copy of the left regular module Λ ′ Λ ′ . Consequently, the strongly tilted algebraΛ ′ = eΛe, i.e., the opposite of End Λ ′ (T ′ ), coincides with Λ ′ . By Theorem 4.6, it therefore suffices to check that P <∞ (mod-Λ ′ ) is contravariantly finite in mod-Λ ′ . To confirm this, we observe that the pair (mod-Λ ′ , e ′ ) in turn satisfies the hypotheses 4.1: Indeed, the precyclic vertices of (Q ′ ) op are precisely e 1 , . . . , e r , whence the simple right Λ ′ -modules of infinite projective dimension are among the quotients e i Λ ′ e i J ′ for i ≤ r, and (e − e ′ )Λ ′ e ′ = 0. The right finitistic dimension of e ′ Λ ′ e ′ = e ′ Λe ′ is in turn zero, because all indecomposable projective right e ′ Λe ′ -modules have the same Loewy length; indeed, this follows by duality from the second of our two hypotheses. Consequently, another application of Proposition 6.1 yields contravariant finiteness of P <∞ (mod-Λ ′ ) as required.
The concluding statements are part of Theorem 3.1.
We deduce Theorem D of [13] as a special case. The proof of the following generalization of Proposition 6.2 is immediate from that of the latter. Recall that an Artin algebra Λ is said to be left serial if all indecomposable projective left Λ-modules are uniserial. The algebras which are left and right serial are also called Nakayama algebras. In [6] it was shown that, for any split left serial algebra Λ, the category P <∞ (Λ-mod) is contravariantly finite in Λ-mod. The conclusion actually carries over to the category mod-Λ of right Λ-modules. Namely: Proposition 6.5. Suppose that Λ is a path algebra modulo relations. If Λ is left serial, then P <∞ (Λ-mod) is contravariantly finite in Λ-mod and P <∞ (mod-Λ) is contravariantly finite in mod-Λ.
Proof. The assertion for left modules was proved in [6]. We only address right Λ-modules.
By hypothesis, Λ ≅ KQ I is left serial. In particular, this means that Q is free of double arrows. Without loss of generality, we assume that Q is a connected quiver; we may further assume that it is not a tree, since otherwise Λ has finite global dimension, which renders the contravariant finiteness claim trivial. By left seriality, Q then contains a single oriented cycle such that all off-cycle vertices are prebut not postcyclic. Consequently, any vertex of Q op either belongs to said cycle or else is post-but not precyclic. On letting e be the sum of the precyclic vertices of Q op , i.e., the vertices located on the oriented cycle in the present situation, we thus obtain a Nakayama algebra eΛe. Hence the P <∞ -categories in both eΛe-mod and mod-eΛe are contravariantly finite in the corresponding ambient module categories by [6,Theorem 5.2]. (The latter may alternatively be deduced from the fact that Nakayama algebras have finite representation type.) On combining this with Proposition 6.1, we obtain the claim. 6.2. Applications to Morita contexts. In this subsection we shall see that Morita contexts provide a tool to construct examples of P <∞ -contravariant finiteness and iteration of strong tilting. Recall that a Morita context (over a ground commutative ring K) consists of a sextuple (A, B, M, N, ϕ, ψ), where A and B are K-algebras, M and N are an A-B-and a B-A-bimodule, which we always assume with the same action of K on the left and on the right, and ϕ ∶ M ⊗ B N → A and ψ ∶ N ⊗ A M → B are morphisms of A-A-and B-B-bimodules, respectively, satisfying certain compatibiliity conditions (see [19]) which are exactly the ones that make Λ = A M N B into a K-algebra with the obvious multiplication. Recall that τ A = Im(ϕ) and τ B = Im(ψ) are two-sided ideals of A and B, respectively, called the trace ideals of the Morita context. The Morita contexts in which we are interested have some additional properties. We assume that K is artinian and A, B, M and N are finitely generated as Kmodules. Such a Morita context will be called a basic Morita context of Artin algebras if Λ is basic or, equivalently, if A and B are basic, τ A ⊆ J(A) and τ B ⊆ J(B), where J(−) denotes the Jacobson radical.
Next we should notice that there is an algebra isomorphism Λ ΛeΛ ≅ B τ B , so that these isomorphic algebras have finite global dimension. Moreover, we have an isomorphism Λ ΛeΛ ≅ Λ(1−e) ΛeΛ(1−e) in Λ-mod, which, by the previous paragraph, implies that Λ ΛeΛ has projective dimension ≤ 1 as a left Λ-module. Note also that This is also an exact sequence in Λ-mod, and we have that pdim Λ (Q k ) ≤ 1 since Q k ∈ add( Λ ΛeΛ ), for all k = 0, 1, ..., t. It then follows that pdim Λ ( Λ(1−e) J(Λ)(1−e) ) < ∞, as desired. The final statement clearly follows from assertion 3 since in those examples A-mod allows for unlimited iteration of strong tilting.
The last theorem and its proof have the following consequence for triangular matrix algebras. Corollary 6.7. Let K be a commutative Artinian ring, let A and B be basic Artin K-algebras and let M and N be a finitely generated A-B-bimodule and B-Abimodule, respectively. Suppose that B has finite global dimension and P ∞ (A-mod) is contravariantly finite in A-mod. The following assertions hold: Proof. Assertion 1 is a direct consequence of Theorem 6.6. As for assertion 2, note that if e = 1 0 0 0 then pdim eΓe (eΓ(1 − e)) < ∞ since we have pdim( A M ) < ∞. We will prove that pdim Γ (ΓeΓ(1 − e)) < ∞ and, arguing as in the last paragraph of the proof of Theorem 6.6, we will conclude that the pair (Γ-mod, e) satisfies the blanket hypotheses of Setting 4.1, and the result will follow from Theorem 4.6 and Corollary 5.6. Note that 0 = (1 − e)Γe and so Γe = eΓe is projective as right eΓe-module. Moreover we have a commutative diagram of K-modules where the upper horizontal arrow is the canonical isomorphism and the lower horizontal one is the multiplication map. It follows that this latter arrow is an isomorphism, which implies that pdim( Γ ΓeΓ(1 − e)) < ∞ since eΓe eΓ(1 − e) has finite projective dimension and the functor Γe ⊗ eΓe − ∶ eΓe-mod → Γ-mod is exact and takes projectives to projectives.
We end the paper by giving non-triangular examples to which Theorem 6.6 applies. We start with the following elementary observation. In the situations (a) and (b), the associated algebra Λ = A M N B satisfies that P <∞ (Λ-mod) is contravariantly finite in Λ-mod (resp., Λ-mod allows for unlimited iteration of strong tilting) if, and only if, so does the algebra A.
Proof. In situation (a), we have an isomorphism of left A-modules M ≅ ⊕ i∈I Ae ti i , where I is the set of i ∈ {1, ..., m} such that the simple left A-module Ae i J(A)e i embeds in top( A M ) and t i > 0 for all i ∈ I. It follows that N ⊗ A M = 0 and the existence of the mentioned Morita context follows by Remark 6.8. Hence in both situations M is projective as a left A-module and the map ψ ∶ N ⊗ A M → B is a monomorphism. In (a) we have that τ B = 0 and in (b) the quiver of B τ B has no oriented cycles. Therefore gl.dim(B τ B ) < ∞ in both cases and assertions 1 and 2 are direct consequences of Theorem 6.6.
When gldim(B) < ∞, the subcategory P <∞ (Λ-mod) is contravariantly finite in Λ-mod (resp. Λ-mod allows for arbitrary iteration of strong tilting) if, and only if, the corresponding property is true for the algebra A. We are thus in the situation of Example 6.9(a), and the conclusions follow from that example. 6.3. A specific path algebra modulo relations. The final example is a nonmonomial path algebra modulo relations whose category of left modules allows for unlimited iteration of strong tilting. Our reduction technique renders verification of this fact significantly less labor-intensive. Example 6.11. Let Λ = KQ I be a specimen of the following class of finite dimensional algebras over a field K, which depends on 4 parameters c 1 , . . . , c 4 ∈ K * . The quiver Q is and I ⊆ KQ is the ideal generated by γα−c 1 δβ, γβ −c 2 δα, αρ−c 3 βσ, and αρ−c 4 τ ν, next to monomial relations which are apparent from the graphs of the indecomposable projective left Λ-modules: show that all simple right Λ ′ -modules embed into the right socle of Λ ′ . Consequently, Theorem 4.6 yields contravariant finiteness of P <∞ (Λ-mod) in Λ-mod.
It is not difficult to directly ascertain that P <∞ (mod-Λ ′ ) is in turn contravariantly finite in mod-Λ ′ , but another application of Theorem 4.6 cuts this task short: Setting e ′ = e ′ 2 + e ′ 3 , where the e ′ i are the primitive idempotents of Λ ′ corresponding to the vertices of Q ′ , it is effortless to check that e ′ satisfies the conditions of Setting 4.1 relative to mod-Λ ′ , and that r.findim e ′ Λ ′ e ′ = 0. Hence P <∞ (mod-e ′ Λ ′ e ′ ) is contravariantly finite in mod-e ′ Λ ′ e ′ , and consequently so is P <∞ (mod-Λ ′ ) in mod-Λ ′ . By Theorem 3.1, we thus conclude that Λ ′ -mod allows for unlimited iteration of strong tilting. This completes the argument.